The conditional expectation of \(\mu _{1i}\)  given \(\mu _{2i}\)  can be derived as
\(E(\mu _{1i}|\mu _{2i} = y) = \mu _1 + \dfrac{\tau _{12}}{\tau _2^2}(y - \mu _2),\) 
which motivates the definition of SROC
[1]}, [2]} as follows.
Let \(x\)  be FPR
(\(1 - \mathrm {specificity}\) ),
SROC is defined by
\(SROC(x; \mu , \Omega )= \mathrm {logit}^{-1} \left[ \mu _1 - \dfrac{\tau _{12}}{\tau _2^2}\lbrace \mathrm {logit}(x)+\mu _2\rbrace  \right].\nonumber \) 
